| L(s) = 1 | + (−1.17 + 0.788i)2-s + (−0.659 − 1.60i)3-s + (0.755 − 1.85i)4-s + (−0.951 − 1.42i)5-s + (2.03 + 1.35i)6-s + (−0.304 + 0.735i)7-s + (0.573 + 2.76i)8-s + (−2.12 + 2.11i)9-s + (2.23 + 0.920i)10-s + (−5.27 − 1.05i)11-s + (−3.46 + 0.0115i)12-s + (−2.38 − 1.59i)13-s + (−0.222 − 1.10i)14-s + (−1.65 + 2.46i)15-s + (−2.85 − 2.79i)16-s + (3.60 + 3.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.830 + 0.557i)2-s + (−0.380 − 0.924i)3-s + (0.377 − 0.925i)4-s + (−0.425 − 0.636i)5-s + (0.831 + 0.554i)6-s + (−0.115 + 0.278i)7-s + (0.202 + 0.979i)8-s + (−0.709 + 0.704i)9-s + (0.708 + 0.291i)10-s + (−1.59 − 0.316i)11-s + (−0.999 + 0.00334i)12-s + (−0.661 − 0.441i)13-s + (−0.0594 − 0.295i)14-s + (−0.426 + 0.635i)15-s + (−0.714 − 0.699i)16-s + (0.874 + 0.874i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 + 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0702817 - 0.272600i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0702817 - 0.272600i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.17 - 0.788i)T \) |
| 3 | \( 1 + (0.659 + 1.60i)T \) |
| good | 5 | \( 1 + (0.951 + 1.42i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (0.304 - 0.735i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (5.27 + 1.05i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (2.38 + 1.59i)T + (4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-3.60 - 3.60i)T + 17iT^{2} \) |
| 19 | \( 1 + (2.23 + 1.49i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (2.89 + 6.98i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-0.806 - 4.05i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + 6.95T + 31T^{2} \) |
| 37 | \( 1 + (2.51 + 3.76i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-1.08 + 0.450i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-1.84 - 0.367i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-4.93 + 4.93i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.88 + 9.47i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (2.08 - 1.39i)T + (22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (0.211 + 1.06i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (-13.8 + 2.75i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (4.58 + 1.89i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-0.638 + 0.264i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-2.62 + 2.62i)T - 79iT^{2} \) |
| 83 | \( 1 + (-1.68 + 2.52i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (14.6 + 6.06i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 0.149iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.39286331472691085033310447873, −10.91484383876511279549665120890, −10.28057810218588344162211805936, −8.636489635907361343988777189609, −8.102486969107109730440043387013, −7.19792590908540707517759931351, −5.88369977782638171304268458518, −5.08189552519601333585562479180, −2.35884333870554517292006213356, −0.31719642614617921838957140282,
2.74839555879916699752227916626, 3.92780744099765866821150700358, 5.42387966817538165138755533392, 7.15903084136817754114992260501, 7.895319423346075925463869244827, 9.383948352652371973414943548980, 10.07707264031912430938004950921, 10.81880130767805763112780384142, 11.63985205980764977565483115428, 12.52648528986709688962253503396
